Galaxy formation is a core theme of cosmology. Galaxies provide the beacons with which we measure the expansion and acceleration of the universe. If they are biased tracers of the underlying matter distribution then progress in cosmology becomes difficult. And there is no question that they are biased tracers: for example we only observe about half of the baryons in the universe in galaxies, the conversion of baryons into stars peaks at about 3 %, without feedback star formation rates are unacceptably early and high, and we have at best a fragmentary understanding of feedback. And our understanding of these issues, and others, is based on relatively local observations, with no fundamental theory nor any real grasp of how the relevant physical processes may vary in extreme conditions such as those encountered in the early universe or near supermassive black holes. Nor have we mentioned the initial mass function of stars or the complex interplay between the fuel for forming stars, the gas reservoirs, and galaxy formation.
However we cannot abandon the search for a theory of galaxy formation, if only because an improved understanding of this subject is central to essentially all major telescope projects, under construction or being planned for the future. To be fair, there has been one enormous advance since the pioneering study of Lifschitz that may be said to have triggered the entire field of how galaxies formed from primordial density perturbations. This has been the detection of cosmic microwave background temperature fluctuations in their damped acoustic oscillations, now beautifully confirmed over many harmonics, which provide the evidence for the seed fluctuations. The physical effect was predicted soon after the discovery of the cosmic microwave background radiation, and evaluated quantitatively for the current cold dark matter-dominated cosmology by solving the Boltzmann equation for the photon coupling with matter. Armed with initial conditions for the primordial density fluctuations, one can now successfully account for the large-scale structure of the universe. However it is the smaller scale astrophysics that is still poorly understood. This review will begin with star formation, and then discuss various aspects of galaxy formation that are beginning to be probed at low and high redshift.